Finite Group Databasebeta

← #11777735

Subgroups — #11777735

Order 1152.

367 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#3716C211yesmaximal, characteristic
2#10590337C211yesmaximal, characteristic
3#10590254C211yesmaximal, characteristic, complemented
4#10589322C211yesmaximal, characteristic, complemented
5C3×#4459C211yesmaximal, characteristic
6C3×#4455C211yesmaximal, characteristic, complemented
7#10589099C211yesmaximal, characteristic, complemented
8#68159—31nomaximal
9#68088—31nomaximal
10C4×#634C2211yescharacteristic
11C3×C4×D12C2211yescharacteristic
12C2×#3716C411yescharacteristic
13C2×#3711—21no—
14C32×C4×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊C8)C2211yescharacteristic
17C3×C4×C3⋊C8C2211yescharacteristic
18C3×C4×C3⋊C8C2211yescharacteristic
19#63573C411yescharacteristic
20#63572—21no—
21#63567C411yescharacteristic
22#63563C411yescharacteristic
23(C2×D3)⋊(C3⋊C8)—21no—
24#63553—21no—
25C4×#214—31no—
26#5012—31no—
27#4830—31no—
28C4×#204—31no—
29#4688—31no—
30C8×Dic6—31no—
31C3×D4⋊C8—31no—
32C3×#322—31no—
33#4479—31no—
34#4459D311yescharacteristic
35#4458—31no—
36#4456—31no—
37#4456—31no—
38#4456—61no—
39#4455D311yescharacteristic, complemented
40C2×#634C2×C411yescharacteristic
41C2×C3×D12C2×C411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C32×C8—21no—
44C32×C42C2311yescharacteristic
45C4⋊(C3⋊Dic3)C2×C411yescharacteristic
46#3750—21no—
47C4×C3⋊Dic3—21no—
48C2×C3⋊(C3⋊C8)—21no—
49C3×(C2×D3)⋊C4—21no—
50C3×C4⋊Dic3C2×C411yescharacteristic
51C2×C3×C3⋊C8—21no—
52C2×C3×C3⋊C8C2×C411yescharacteristic
53C2×C3×C3⋊C8C2×C411yescharacteristic
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3716—22no—
57#3711—42no—
58#1588—91noSylow
59C3×C4×D4—31no—
60C2×#214—31no—
61C2×#208—61no—
62C2×#204—31no—
63C2×ΓL(1,𝔽25)—61no—
64C4×D12C2×D311yescharacteristic
65C4×Dic6—31no—
66C4×Dic6—31no—
67C4×Dic6—61no—
68C3×C4⋊C8—31no—
69C3×C4×C8—21no—
70C3×C4×C8—31no—
71C3×C4×C8—31no—
72C3×C4×C8C2×D311yescharacteristic
73C3×C4×C8C2×D311yescharacteristic
74#832—31no—
75#830—61no—
76#821—31no—
77(C2×D3)⋊C8—61no—
78#817—31no—
79#816—31no—
80Dic3⋊C8—61no—
81Dic3⋊C8—61no—
82C8×Dic3—61no—
83C8×Dic3—61no—
84#810—31no—
85#808—31no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—61no—
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4×C3⋊C8C2×D311yescharacteristic
91C4×C3⋊C8C2×D311yescharacteristic
92C22×C3×D3—21no—
93C2×C32×C4C22×C411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4#4111yescharacteristic
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98#634—21no—
99#634M1612yescomplemented
100C2×C3×Dic3—21no—
101C3×D12—21no—
102C3×D12—21no—
103C3×C4×D3—41no—
104C3×C4×D3—41no—
105C32×C8—22no—
106C3⋊(C3⋊C8)—41no—
107C3×C3⋊C8—21no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112C8×Q8—91no—
113C4×SD16—91no—
114#436—91no—
115#335—91no—
116Q8⋊C8—91no—
117D4⋊C8—31no—
118#322—31no—
119C2×C3×D4—31no—
120C22×C3×C4—61no—
121C2×D12C4×D311yescharacteristic
122C2×C4×D3—21no—
123C2×Dic6—31no—
124C2×Dic6—31no—
125C2×Dic6—61no—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—31no—
129C2×C3×C8—31no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—41no—
133C2×C3×C8—61no—
134C3×C4⋊C4—31no—
135C3×C22⋊C4—61no—
136C3×C42—21no—
137C3×C42C22×D311yescharacteristic
138C3×C42C22×D311yescharacteristic
139#214—62no—
140(C2×D3)⋊C4—21no—
141C4⋊Dic3—31no—
142C4⋊Dic3—31no—
143C4⋊Dic3—61no—
144C4⋊Dic3C4×D311yescharacteristic
145Dic3⋊C4—61no—
146Dic3⋊C4—61no—
147Dic3⋊C4—121no—
148C4×Dic3—61no—
149C4×Dic3—61no—
150C4×Dic3—121no—
151#208—122no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—121no—
156C2×C3⋊C8C4×D311yescharacteristic
157C2×C3⋊C8C4×D311yescharacteristic
158C2×C3⋊C8D1211yescharacteristic
159C2×C3⋊C8D1211yescharacteristic
160#204—62no—
161ΓL(1,𝔽25)—122no—
162C22×C32C4×D411yescharacteristic
163C2×C3×D3—41no—
164C2×C3×D3—41no—
165C2×C3×D3—41no—
166C32×C4—21no—
167C32×C4C2×M1611yescharacteristic
168C32×C4#13111yescharacteristic
169C32×C4AGL(1,ℤ/8)11yescharacteristic
170C32×C4#13711yescharacteristic
171C3⋊Dic3—22no—
172C3⋊Dic3—41no—
173C3×Dic3—41no—
174C2×SD16—91no—
175C2×M16—181no—
176C4×Q8—91no—
177C4×D4—31no—
178#106—91no—
179C4⋊C8—31no—
180C4⋊C8—91no—
181C4⋊C8—181no—
182#103—91no—
183#102—91no—
184C22⋊C8—181no—
185C4×C8—31no—
186C4×C8—31no—
187C4×C8—181no—
188C4×C8D3211yescharacteristic
189C23×C3—61no—
190C22×D3—21no—
191C3×D4—61no—
192C3×D4—61no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—61no—
197C2×C3×C4—121no—
198C2×C3×C4—121no—
199C2×C3×C4C2×C4×D311yescharacteristic
200C2×C3×C4C2×C4×D311yescharacteristic
201C2×C3×C4C2×D1211yescharacteristic
202C2×C3×C4#23511yescharacteristic
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4#23711yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—121no—
211C2×Dic3—121no—
212D12—21no—
213D12—21no—
214C4×D3—41no—
215C4×D3—41no—
216Dic6—32no—
217Dic6—32no—
218Dic6—62no—
219Dic6—61no—
220Dic6—61no—
221Dic6—121no—
222C3×C8—22no—
223C3×C8—22no—
224C3×C8—42no—
225C3×C8—61no—
226C3×C8—61no—
227C3×C8—61no—
228C3×C8—61no—
229C3×C8—121no—
230C3⋊C8—21no—
231C3⋊C8—21no—
232C3⋊C8—21no—
233C3⋊C8—21no—
234C3⋊C8—41no—
235C3⋊C8—121no—
236C3⋊C8—121no—
237C3⋊C8—241no—
238C2×C32#43511yescharacteristic
239C2×C32#44011yescharacteristic
240C2×C32#44311yescharacteristic
241C3×D3—81no—
242C2×Q8—91no—
243C2×D4—31no—
244C22×C4—61no—
245SD16—182no—
246M16—362no—
247C2×C8—21no—
248C2×C8—31no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—61no—
253C2×C8—181no—
254C4⋊C4—31no—
255C4⋊C4—91no—
256C4⋊C4—181no—
257C22⋊C4—61no—
258C42—181no—
259C42C2×D3211yescharacteristic
260C22×C3—21no—
261C22×C3—121no—
262C22×C3—121no—
263C22×C3—121no—
264C22×C3C4×D1211yescharacteristic
265C22×C3#67⋊C411yescharacteristic
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—121no—
277C3×C4#90111yescharacteristic
278C3×C4D3×M1611yescharacteristic
279C3×C4D3×M1611yescharacteristic
280C3×C4#90711yescharacteristic
281C3×C4#90811yescharacteristic
282C3×C4#90911yescharacteristic
283C3×C4#91111yescharacteristic
284C3×C4#91511yescharacteristic
285Dic3—41no—
286Dic3—62no—
287Dic3—62no—
288Dic3—122no—
289Dic3—121no—
290Dic3—121no—
291Dic3—241no—
292C32#158811yescharacteristic, complemented, Sylow
293C23—61no—
294Q8—92no—
295Q8—181no—
296D4—61no—
297D4—61no—
298C2×C4—61no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—181no—
302C2×C4—181no—
303C2×C4C4×D3211yescharacteristic
304C2×C4D3×D1211yescharacteristic
305C2×C4#380411yescharacteristic
306C8—22no—
307C8—61no—
308C8—61no—
309C8—61no—
310C8—61no—
311C8—121no—
312C8—361no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—241no—
317C2×C3#470011yescharacteristic
318C2×C3#470611yescharacteristic
319C2×C3#471011yescharacteristic
320C2×C3#472811yescharacteristic
321C2×C3#475711yescharacteristic
322C2×C3#482411yescharacteristic
323D3—81no—
324C22—121no—
325C22—121no—
326C22—121no—
327C22#6388011yescharacteristic
328C4—21no—
329C4—121no—
330C4—182no—
331C4—361no—
332C4#6379111yescharacteristic
333C4#6379511yescharacteristic
334C4#6380011yescharacteristic
335C4#6380411yescharacteristic
336C3—21no—
337C3#6808811yescharacteristic, complemented
338C3#6815911yescharacteristic, complemented
339C2—241no—
340C2#1059034111yescharacteristic
341C2#1059073911yescharacteristic
342C2#1059074711yescharacteristic
subgroupG/H
centerC2×C4D3×D12
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4×C8C22
socleC22×C32C4×D4
prime radicalC2×C3×D12C2×C4
cosocleC32×C42C23
hypercenterC4×C8D32
hypocenterC32#1588
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #10590254C2semidirect
4. #10589322C2semidirect
6. C3×#4455C2semidirect
7. #10589099C2semidirect
39. #4455D3semidirect
99. #634M16semidirect
292. C32#1588semidirect
337. C3#68088semidirect
338. C3#68159semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×#37162
#105903372
#105902542
#105893222
C3×#44592
C3×#44552
#105890992
#681593
#680883